chapter3.page02 - dene matrix addition, multiplication, and...

2 1. BRIEF INTRODUCTION TO VECTORS AND MATRICES other elements are 0. So 2 × 2 and 3 × 3 zero matrices are • 1 0 0 1 ‚ and   1 0 0 0 1 0 0 0 1   A vector is a matrix with one row or one column. In this chapter, a vector is always a matrix with one column as • x 1 x 2 ‚ for a two-dimensional vector and   x 1 x 2 x 3   for a three dimensional vector. Here the element has only one index that denotes the row position (Sometimes we use diﬀerent variable to denote number in diﬀerent position such as using • x y ‚ for a 2-dimensional vector). We use bold lower case, such as v , to denote a vector. 1.2. Operations on Matrices. Arrange number in rectangular fashion, as a matrix, itself is not something terribly interesting. The most important advantage from that kind arrangement is that we can

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Unformatted text preview: dene matrix addition, multiplication, and scalar multiplication. Definition 1.1 . (i) Equality: Two matrix A = ( a ij ) and B = ( b ij ) are equal if corresponding elements are equal, i.e. a ij = b ij . (ii) Addition: If A = ( a ij ) and B = ( b ij ) and the sum of A and B is A + B = ( c ij ) = a ij + b ij . (iii) Scalar Product: If A = ( a ij ) is matrix and k is num-ber(scalar), the kA = ( ka ij ) is product of k and A . From the above denition, we see that, to multiply a matrix by a number k , we simply multiply each of its entries by k ; to add two matrices we just add their corresponding entries; A-B = A +(-1) B . Example 1.1 . Let A = 2 3-1 4...
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